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In 1966, William Russell Abel, Ansel Cochran Anderson and John Charles Wheatley sent acoustic waves through bulk liquid helium-3 at $0.32\mathrm{atm}$ and frequencies $15.4$ and $45.5\mathrm{MHz}$. A quartz delay line and receiving acoustic transducer bounded a $0.138\mathrm{mm}$ liquid gap; shielding reduced electrical pickup. Powdered cerium magnesium nitrate provided magnetic thermometry during a slow warm-up. Pulse amplitudes measured acoustic attenuation; phase difference comparison measured velocity. Calibration pulses checked gain; early received cycles avoided multiple reflections. On cooling through a maximum, acoustic attenuation changed from $\alpha \propto {\omega }^2/T^{\ast 2}$ to $\alpha \propto T^{\ast 2}$ with a velocity shift supporting zero sound; $T^{\ast }$ is their magnetic temperature scale. Weak acoustic attenuation was harder to resolve.^[ref]

This 1966 test followed Lev Landau's 1962 Nobel Prize in Physics.^[ref]

The experiment concerned normal liquid helium-3, distinct from the later discovery of superfluid helium-3.^[ref] Landau's original papers are theoretical calculations, not reports of an apparatus.

An experimental condensed matter physicist who coauthored an observation of zero sound in helium-3. The experiment probed a collective density wave in a regime where quasiparticle collisions are slow compared with the oscillation. This distinguishes collisionless quantum-fluid dynamics from ordinary hydrodynamic sound.^[ref]

The short-wavelength root of the degenerate-plasma dielectric response, approaching the edge of the occupied parallel velocity interval from outside. Writing $s=1+\delta$ makes its logarithmic singularity give the exponentially small positive correction in the title. It is the high-wavenumber limit of the charged Langmuir wave branch in this approximation; neutral zero sound has a different interaction and long-wavelength limit.

Why can a strongly interacting liquid resemble a gas? Its low-energy excitations can be long-lived Landau quasiparticles near a Fermi surface. Normal liquid helium-3 motivates this Fermi liquid description: interactions survive in effective excitation energies and interactions.^[ref]

Interacting one-dimensional fermion systems often form Luttinger liquids; non-Fermi liquids also occur in higher dimensions. A Cooper instability can produce a paired phase.^[ref]

The Fermi-surface renormalization-group scaling below organizes the low-energy interactions; zero sound tests whether their collective dynamics survives even when collisions are rare.

An ordinary acoustic wave involving a pressure and density oscillation in local thermodynamic equilibrium. In a Fermi liquid, this hydrodynamic response requires collisions fast compared with the oscillation period; zero sound instead describes a collisionless collective mode.

Angular-relaxation collision time $\tau$ distinguishes first sound ($\omega \tau \ll 1$) from zero sound ($\omega \tau \gg 1$). Collisions establish local thermodynamic equilibrium in the former.^[ref]

For the figure's stipulated law $\omega \tau =(T/T_{\times }{)}^{-2}$, cooling from $T=5T_{\times }$ to $T=T_{\times }/5$ changes $\omega \tau$ from $0.04$ to $25$, without changing the driving angular frequency. The crossover temperature is defined by $\omega \tau (T_{\times })=1$. Calling zero sound a high-frequency mode therefore means high relative to the collision rate $1/\tau$, while its excitation energy must still remain within the low-energy domain of the Fermi liquid.

Consider a neutral, isotropic, unpolarized three-dimensional spin-one-half Fermi liquid, with only $F_0^s>0$ nonzero. A longitudinal density wave with angular frequency $\omega$ and wavevector magnitude $q$ has $s=\omega /(q{v}_F)=c_0/v_F$. The collisionless angular equation for the energy displacement $\nu (u)$ of the Fermi surface isAveraging givesThe bracket decreases from infinity to zero on $1<s<\infty$: each $F_0^s>0$ gives one undamped root above the small-$q$ particle-hole continuum.^[ref]

For an original example, $F_0^s=2$ gives $s=1.16474145$. Since $F_1^s=0$ here, the Principle of Galilean relativity gives $m^{\ast }=m$ and taking the hydrodynamic limit before $T\to 0$ gives $c_1/v_F=\sqrt{(1+F_0^s)/3}=1$. Stipulating $v_F=50\mathrm{m}{\mathrm{s}}^{-1}$ gives $c_0\simeq 58.24\mathrm{m}{\mathrm{s}}^{-1}$ and $c_1=50\mathrm{m}{\mathrm{s}}^{-1}$. These are model choices, not helium-3 measurements. At large $F_0^s$, expanding the logarithm gives $s\sim \sqrt{F_0^s/3}$; for small positive $F_0^s$, the root approaches $1$ exponentially closely, so numerical resolution matters. A charged electron liquid needs its long-range interaction included; this neutral model is not its long-wavelength density mode.

An experimental physicist who coauthored measurements of zero sound in liquid helium-3. At sufficiently low temperature, collisions become too infrequent to maintain ordinary local equilibrium during an acoustic cycle. The resulting collective response tests the theory of a Fermi liquid through propagation of a density disturbance.^[ref]

The particle-hole continuum is the range of energy and momentum transfers that can excite an occupied fermion state into an unoccupied state, leaving a hole behind. For a free gas at zero temperature, choose $|\mathbf{k}|<k_F$ and $|\mathbf{k}+\mathbf{q}|>k_F$; the excitation energy is ${ϵ}_{\mathbf{k}+\mathbf{q}}-{ϵ}_{\mathbf{k}}$.

Varying the initial momentum produces a continuum at fixed transferred wavevector. For small $q$ in an isotropic gas, its positive frequencies extend approximately up to $v_{F}q$. A collective mode overlapping these excitations can decay by creating particle–hole pairs; a zero sound branch outside that region can avoid this particular damping channel. Interactions change the response and effective parameters, so the free-gas boundary is a useful starting model rather than a universal exact spectrum.

Landau quasiparticles describe the long-lived low-energy excitations of an interacting normal Fermi liquid. Starting from a free Fermi gas, imagine switching on interactions without changing the phase or destroying the one-to-one labelling of low-energy states. The resulting excitation carries the quantum numbers of a dressed particle but has a renormalized energy and interactions with other excitations. Its microscopic wavefunction need not resemble a bare electron state. The occupied momenta define a Fermi surface, and the quasiparticle energy measured from the chemical potential vanishes on that surface.

The Landau quasiparticle energy functional expresses the change near equilibrium in terms of occupation changes:Differentiation gives the energy of one quasiparticle in the distorted distribution. This is a phenomenological expansion around the true interacting ground state, not the assertion that its total energy equals that of independent particles. For an isotropic system, spin-symmetric and spin-antisymmetric components of $f$ are expanded in Legendre polynomials. Multiplication by the appropriate Fermi-surface density of states defines dimensionless Landau parameters $F_l^s,F_l^a$. The original formulation relates a change in the distribution to the energy shift.^[ref]

Pauli blocking makes the low-energy excitations sharp. The phase space for a quasiparticle to scatter while creating another particle–hole pair shrinks as the Fermi surface is approached, giving the regular three-dimensional low-energy Fermi-liquid quasiparticle lifetimeThus at zero temperature its width is small compared with its excitation energy. The single-particle spectral function has a pole with quasiparticle residue $Z$, generally less than one, and an incoherent remainder. In a conventional Fermi liquid the occupation distribution has a discontinuity of height $Z$ at the Fermi surface, rather than a unit free-particle step.

For an isotropic three-dimensional system, define $N^{\ast }(0)$ to include both spins. The quasiparticle density of states determines the low-temperature electronic heat capacity, $C_V/V=({\pi }^2/3)k_B^2{N}^{\ast }(0)T$. In the usual Landau-parameter normalization, the spin response is modified by $1/(1+F_0^a)$ and the compressibility by $1/(1+F_0^s)$. In a Galilean-invariant continuum, $m^{\ast }/m=1+F_1^s/3$; that particular relation must not be imposed unchanged on a crystal band. A denominator approaching zero signals a possible instability of the assumed normal state.

The same interaction functional and a kinetic equation describe collective zero sound when the excitation frequency exceeds the collision rate. Ordinary first sound is the collision-dominated hydrodynamic limit. The theory organizes thermodynamics, response and transport through a small set of measured effective parameters while explaining why interactions coexist with long-lived fermionic excitations. It is not applicable without qualification in phases where the Fermi-liquid quasiparticle pole disappears or a gap opens over the relevant Fermi surface.

A normal Fermi liquid describes an interacting fermion system whose low-energy excitations remain continuously connected to adding particles to or removing particles from a Fermi gas. The elementary long-lived excitation is a Landau quasiparticle: adding a particle also changes the surrounding many-body state. Its energy, velocity and interactions are renormalized, even though its electric charge and spin remain those appropriate to the excitation.

Near an isotropic Fermi surface,Residual interactions make the quasiparticle energy depend on the occupations of other quasiparticles, expressed by the Landau quasiparticle energy functional. Its angular components give the Landau parameters controlling density and spin responses. Thus the successful particle picture does not mean that interactions have become zero.

Pauli blocking restricts the final states available for low-energy collisions. In an ordinary three-dimensional Fermi liquid with regular interactions, a representative decay law isAt $T=0$, the width vanishes faster than $|\xi |$, so an excitation becomes increasingly sharp as it approaches the Fermi surface. Its coherent spectral weight is the quasiparticle residue $Z$, and the physical electron occupation has a discontinuity of height $Z$ at zero temperature. Incoherent spectral weight accounts for the remainder. The quasiparticle occupations themselves obey a Fermi-Dirac distribution.

Several leading thermodynamic signatures retain their gas-like form but have interaction-dependent coefficients. In particular,for an isotropic spin-one-half model, with $N^{\ast }(0)$ the spin-summed quasiparticle density of states per volume. effective mass changes $N^{\ast }(0)$, while the dimensionless interaction parameters $F_0^a,F_0^s$ separately change the spin susceptibility and compressibility. Collisionless collective density motion can propagate as zero sound, distinct from the single-particle excitations.

The $T^2$ quasiparticle scattering rate often underlies a low-temperature $T^2$ contribution to metallic electrical resistivity, but a mechanism relaxing electrical electric current is needed. In a clean Galilean-invariant continuum, electron-electron collisions conserve total momentum and do not alone give a finite dc electrical resistivity. Crystal Umklapp scattering or other current-relaxing processes change that conclusion.

Normal liquid helium-3 and many metals motivate the Fermi-liquid framework. Pairing, magnetic ordering or other transitions can reconstruct the low-energy states. Generic interacting one-dimensional fermion systems instead form a different low-energy state, so the three-dimensional quasiparticle conclusions should not simply be transferred to a chain.

A normal Fermi liquid has interacting fermions whose low-energy excitations can be followed continuously from those of a Fermi gas, provided no phase transition destroys that correspondence. A Landau quasiparticle is a particle together with the surrounding many-body response. It has the appropriate charge and spin, a renormalized dispersion and a long lifetime close to the Fermi surface. Interactions remain in both the quasiparticle parameters and the interactions between quasiparticles; they have not been discarded.

Near an isotropic Fermi surface,The last relation defines the local isotropic Fermi-liquid effective mass. The Landau quasiparticle energy functional expresses how changing occupations shifts other quasiparticle energies:with normalization of the sum absorbed into $f$. Its angular components give Landau parameters describing collective density and spin response.

The Pauli exclusion principle severely restricts the available final states for low-energy collisions. In an ordinary three-dimensional Fermi liquid with regular interactions,At zero temperature the decay width decreases faster than $|\xi |$, making quasiparticles increasingly sharp near the Fermi surface. Their coherent spectral weight is the quasiparticle residue $0<Z\le 1$, and the true electron momentum distribution has a discontinuity of height $Z$ at zero temperature. The rest of the spectral weight is incoherent. The quasiparticle occupation itself still follows a Fermi-Dirac distribution.

The low-temperature electronic heat capacity is linear, $C_V/\mathcal{V}=({\pi }^2/3)k_B^2{N}^{\ast }(0)T$, where $N^{\ast }(0)$ is the spin-summed quasiparticle density of states. In an isotropic spin-one-half model, the uniform spin susceptibility and compressibility take the formsThus enhanced effective mass and residual interactions can separately change observable responses. The familiar $T^2$ electrical resistivity requires a mechanism relaxing electrical current, such as allowed Umklapp processes; collisions in a perfectly clean Galilean-invariant continuum conserve total momentum and do not alone produce a finite dc resistivity.

Zero sound is a collisionless collective density oscillation sustained by the quasiparticle interaction; ordinary first sound occurs when collisions establish local equilibrium during a cycle. Metals and normal liquid helium-3 supply important examples. The description fails in phases with no sufficiently sharp low-energy quasiparticles, such as generic interacting one-dimensional systems, and must be modified after pairing or another ordering transition.

An experimental physicist who coauthored the 1966 zero sound propagation experiment with Ansel Cochran Anderson and John Charles Wheatley. Sound measurements in very cold helium-3 tested whether collective density motion survives in the collisionless regime predicted for a Fermi liquid.^[ref]

Zero sound is a collisionless collective Fermi surface deformation sustained by Landau quasiparticle interactions.^[ref]

A useful analogy is a crowd whose members respond to their neighbors' density without repeatedly bumping into them: the restoring interaction can remain effective when collisions cannot relax the distribution within one cycle. Zero sound is collective; it is not a single Landau quasiparticle outrunning every other excitation.